Insights into Antisite Defect Complex Induced High Ferro-Piezoelectric Properties in KNbO3 Perovskite: First-Principles Study

Improving ferro-piezoelectric properties of niobate-based perovskites is highly desirable for developing eco-friendly high-performance sensors and actuators. Although electro-strain coupling is usually obtained by constructing multiphase boundaries via complex chemical compositions, defect engineering can also create opportunities for novel property and functionality advancements. In this work, a representative tetragonal niobate-based perovskite, i.e., KNbO3, is studied by using first-principles calculations. Two intrinsic types of Nb antisite defect complexes are selected to mimic alkali-deficiency induced excess Nb antisites in experiments. The formation energy, electronic profiles, polarization, and piezoelectric constants are systematically analyzed. It is shown that the structural distortion and chemical heterogeneity around the energetically favorable antisite pair defects, i.e., (NbK4·+KNb4′), lower the crystal symmetry of KNbO3 from tetragonal to triclinic phase, and facilitate polarization emergence and reorientation to substantially enhance intrinsic ferro-piezoelectricity (i.e., spontaneous polarization Ps of 68.2 μC/cm2 and piezoelectric strain constant d33 of 228.3 pC/N) without complicated doping and alloying.

Meanwhile, structural point defects, i.e., vacancies, dopants, and interstitials, are decisively important for electronic and electrical functionalities of ferroelectric perovskites.In particular, defect complexes, consisting of clusters or aggregates of point defects, play a vital role in ferro-piezoelectric, electro-strain, and electromechanical properties of ferroelectric crystalline materials.For instance, large electro-strain has been generally observed in various lead-free ferroelectric perovskites, which stems from a reversible domain-switching mechanism as a result of energetically favorably formed defect complexes involving vacancies and acceptor/donor dopants on either A-or B-site cations [22][23][24][25].Furthermore, the coupling between charged hetero-valence dopants and vacancies can generate electric and elastic dipole moments that significantly influence the polarization switching and domain wall motion, thereby giving rise to ferroelectric "hardening" or "softening" depending on the type of doping (i.e., acceptor or donor dopants) [26][27][28].Whereas these dopants are usually exotic substituents or impurity atoms, antisite point defects and associated defect complexes can be intrinsically formed due to deviations from ideal stoichiometry in chemically simple, lead-free ferroelectrics.Recently, a giant effective piezoelectric coefficient (d *   33 ) with a high Curie temperature has been achieved in sputtering deposited alkali-deficient niobate-based perovskite (NaNbO 3 and (K, Na)NbO 3 ) epitaxial thin films, which contain self-assembled antisite or planar defect complexes [29][30][31][32][33].The formation of excess Nb antisites, accompanied with compensated A-site vacancies or antisites, becomes energetically favorable in these tetragonal niobate perovskites, and their assembly can result in a variety of strain and polar states (i.e., antisite nanopillars and planar faults) together with chemical and structural heterogeneity, which are responsible for the superior electromechanical response.However, a complete and direct picture for underlying principles and mechanisms in relation to the atomic structure of antisite defect complexes is still an imperative requirement, so that these defects can be deliberately tuned and perovskite crystals with improved ferro-piezoelectric properties can be rationally designed.
Undoubtedly, various theoretical investigations of niobate-based perovskites have been performed using first-principles calculations, but to our knowledge, these studies have mostly focused on isolated point defects [34][35][36][37].Nevertheless, the effects of structural defects on macroscopic ferro-piezoelectric performance should originate from the formation of intricate defect complexes, rather than point defects separately.Therefore, we performed first-principles density functional theory (DFT) calculations of a representative tetragonal niobate-based ferroelectric perovskite model, i.e., KNbO 3 .Two main types of neutral-charged antisite defect complexes, namely, (Nb 4• K + 4V ′ K ) and (Nb 4• K + K 4′ Nb ) in Kröger-Vink notation, were simulated.The effects of the antisite defects on the structural, electronic, ferroelectric, and piezoelectric properties of KNbO 3 single crystals were also thoroughly examined.The ultimate objective of this work is thus to fill the knowledge gap and provide informative and comprehensive understanding of the underlying physical mechanisms of antisite defect complexes induced superior ferro-piezoelectricity in niobate-based perovskites from an electronic/atomic perspective.

Computational and Structural Details
In this work, all first-principles DFT calculations are carried out using the Vienna ab-initio simulation package (VASP) through a plane-wave basis set with the projector augmented wave (PAW) method for the core-valence interactions [38][39][40].The exchangecorrelation energy of electrons is depicted by generalized gradient approximations (GGAs) in the form of the Perdew-Burke-Emzerh (PBE) scheme [41,42].K 3s 2 3p 6 4s 1 , Nb 4p 6 5s 1 4d 4 , and O 2s 2 2p 4 are treated as semi-core states, and are included in the valence.The energy cutoff for the plane-wave basis set is 600 eV.The Brillouin zone integrations are sampled by using a gamma-centered method with 4 × 4 × 3 Monkhorst-Pack k-point grids at a spacing of 0.03 Å −1 .The energy convergence accuracy is chosen at 1.0 × 10 −6 eV, and the interaction force between atoms is kept below 0.01 eV/Å.The conjugate-gradient algorithm is utilized to relax all ions into their instantaneous ground state.spacing of 0.03 Å −1 .The energy convergence accuracy is chosen at 1.0 × 10 −6 eV, and the interaction force between atoms is kept below 0.01 eV/Å.The conjugate-gradient algorithm is utilized to relax all ions into their instantaneous ground state.
To study antisite defect effects, the tetragonal structure of KNbO3 having space group P4mm (no.99) with experimental lattice constants, i.e., the unit cell parameters of La = Lb = 3.996 Å and Lc = 4.063 Å, is used as the initial configuration.The 2 × 2 × 3 supercell of the tetragonal KNbO3 unit cells (60 atoms) is employed to construct the antisite defect models, and their optimized atomic configurations are shown in Figure 1.In this work, we mainly pay attention to the excess Nb antisite defect, which introduces extra +4 charges to the model system.To balance charge and enhance structural stability, Nb antisites with A-site vacancies or antisite, namely (Nb K 4• + 4V K ′ ) and (Nb K 4• + K Nb 4′ ), are possible combinations of defect complexes in KNbO3.For (Nb K 4• + 4V K ′ ), the Nb K 4• is compensated by four K vacancies.In this scenario, a K atom at the A-site is randomly replaced by a Nb atom, and another four K vacancies are introduced to generate KNbO3 with (Nb K 4• + 4V K ′ ) defect clusters, which is simply abbreviated as the Nb K 4• -4V K ′ -KNbO3 model.For the second scenario, the Nb K 4• is directly compensated by a K Nb 4′ antisite pair defect such that an A-site K atom is randomly selected and swapped with a Nb atom at the B-site, giving rise to the Nb K 4• -K Nb 4′ -KNbO3 model.It is noted that different antisite structures could be built according to the locations of these point defects.Hence, only the structure with the lowest total energy after geometry optimization is considered for the two antisite defect complex models in Figure 1b,c.Additionally, the density functional perturbation theory (DFPT) method is employed to calculate and extract elastic and piezoelectric properties as the second derivatives of the total internal energy [43,44] as: where C ij is the elastic stiffness tensor with i,j = {1, 2, 3. ..6}, η is the homogeneous strain, E is the internal total energy, ε is the homogeneous electric field, and e αj is the piezoelectric stress tensor with α = {1, 2, 3}.It is also interesting to note that, among several possible configurations, both the Nb 4• K -4V ′ K -KNbO 3 and Nb 4• K -K 4′ Nb -KNbO 3 models with antisites and vacancy defects located on nearest-neighbor sites possess the lowest energy and thus the most stable atomic structure (see Figure 1 and Tables S1 and S2), indicating that the Nb antisite defect tends to bind strongly to either K vacancies or K antisite defect, which is in good accordance with the DFT predictions for AgNbO 3 with antisite pair defects Nb 4•  Ag and Ag 4′ Nb [47].For the pristine KNbO 3 , the Nb cation is octahedrally coordinated with the O anions forming the corner-shared NbO 6 octahedron, which contains three types of Nb-O bonds (one of them four, and the other two with one each in number) in Figure 1d.The local off-center displacement of Nb cations relative to O anions is revealed by the difference in the Nb-O bond lengths along the c axis (i.e., ∆ = 0.52 Å) and is regarded as the intrinsic origin of the ferroelectricity in tetragonal KNbO 3 .However, in addition to the typical coordination number of 6 for Nb, the two different antisite defect complexes result in NbO 5 and NbO 8 groups with coordination numbers of 5 and 8, respectively, as revealed in Figure 1e,f.The bond lengths of Nb-O bonds formed by the Nb and surrounding O atoms in the three different environments are also shown in Figure 1d-f, where the cutoff of Nb-O bond length is determined from the bond valence parameter method [48][49][50] and is chosen as ~2.453Å.It is clearly observed that the oxygen polyhedron presents significant distortion and tilting after incorporating the two antisite defect clusters, which in turn plays a vital role in electronic and electrical properties of KNbO 3 crystals.Although the tetragonal phase structure is still maintained in Nb 4• K -4V ′ K -KNbO 3 , the in-plane anisotropy is introduced and the out-of-plane bond length variations become larger as compared to the pristine KNbO 3 .This is mainly attributed to the lacking K-O bonds as well as a stronger coupling between Nb and O than that of K and O, thereby leading to the department of the nearest O from V ′ K and the remarkable downward movement of Nb 4• K , which is accompanied by the distortion of oxygen polyhedra.For Nb 4• K -K 4′ Nb -KNbO 3 , the structural asymmetry together with local heterogeneity is further intensified to compensate for the loss of the incipient NbO 6 and KO 12 groups induced by the inversion of neighboring Nb and K cations.Moreover, for the two antisite-defect KNbO 3 models, atomic relaxations in the vicinity of the defects are likely caused, and the positions of the antisite defects, especially Nb 4• K , are displaced substantially along the 001 and/or [110] directions, giving rise to local dipoles and polarization, thus potentially boosting ferroelectric and piezoelectric performance.
To examine the relative difficulty of incorporating the two antisite defect complexes into the lattice, the formation energy of the charge-neutral (Nb 4• K + 4V ′ K ) and (Nb 4• K + K 4′ Nb ) clusters can be calculated as: where E F denotes the defect formation energy, E d t (E 0 t ) is the total energy of the supercell with (without) the defects, µ X is the atomic chemical potential of the element X, and n is the number of formula units included in the supercell.Generally, the smaller the formation energy, the easier it is for antisite defect clusters to enter the lattice position.Consequently, as seen in Table 2, E F is 10.01 and 3.71 eV for (Nb 4• K + 4V ′ K ) and (Nb 4• K + K 4′ Nb ), respectively, signifying that the antisite defect cluster formation is non-spontaneous, and more importantly, the (Nb 4• K + K 4′ Nb ) pair defects are much easier to form than (Nb 4• K + 4V ′ K ).This is mainly attributed to the much greater energy cost required for losing four K cations at the same time than for of replacing only one Nb by K in KNbO 3 crystals.
Table 2. Defect-forming energy E F (eV) and element chemical potential µ X (eV).

Electron Localization Function
The electron localization function (ELF) is employed to analyze variation in the local chemical bonding environment of the pristine and antisite-defect KNO 3 , as shown in Figure 2.After replacing the A-site cation in Figure 2b, the Nb cation tightly bounds to eight O anions (four each at the top and bottom) and its center shifts downward remarkably, leading to a local deformed but yet symmetrical chemical bonding environment in the (110) plane.In addition to Nb 4• K , the surrounding vacant four K atoms help aggravate the twisting and displacement of first-nearest-neighboring O atoms, which finally induces the noticeable distortion of the NbO 6 octahedron.The position of the adjacent Nb cation (i.e., Nb 3 ) is also slightly shifted outward due to the alike charge repulsion of Nb 4• K .However, the B-site (001) plane far away from Nb 4• K displays a highly uniform and similar chemical bonding environment to the pristine KNbO 3 , as seen in Figure 2d,e.In contrast, for the antisite pair defects in Figure 2c, both the Nb 4• K and K 4′ Nb antisites are displaced significantly along the 001 and [110] directions, resulting in asymmetrically structural and chemical bonding changes around the defect cluster.This undoubtedly contributes to the tetragonalto-triclinic phase transition for Nb 4• K -K 4′ Nb -KNbO 3 .More intriguingly, the antisite pair defects produce more electron accumulation on Nb and O atoms on B-site planes (see Figure 2f,g), as compared to KNbO 3 and Nb 4• K -4V ′ K -KNbO 3 .Overall, the distinction of chemical bonding in various sites suggests that the influence of the antisite defect complexes is localized, which is a positive factor in enhancing ferroelectric and piezoelectric properties for defected KNbO 3 .
Materials 2024, 17, x FOR PEER REVIEW 6 of 14 Nb3) is also slightly shifted outward due to the alike charge repulsion of Nb K 4• .However, the B-site (001) plane far away from Nb K 4• displays a highly uniform and similar chemical bonding environment to the pristine KNbO3, as seen in Figure 2d,e.In contrast, for the antisite pair defects in Figure 2c, both the Nb K 4• and K Nb 4′ antisites are displaced significantly along the [001 � ] and [1 � 1 � 0] directions, resulting in asymmetrically structural and chemical bonding changes around the defect cluster.This undoubtedly contributes to the tetragonal-to-triclinic phase transition for Nb K 4• -K Nb 4′ -KNbO3.More intriguingly, the antisite pair defects produce more electron accumulation on Nb and O atoms on B-site planes (see Figure 2f,g), as compared to KNbO3 and Nb K 4• -4V K ′ -KNbO3.Overall, the distinction of chemical bonding in various sites suggests that the influence of the antisite defect complexes is localized, which is a positive factor in enhancing ferroelectric and piezoelectric properties for defected KNbO3.

Band Structure and Density of States
The band structures along the symmetry directions as well as the total and partial electronic density of states (DOS) are further shown in Figure 3. Figure 3a shows that the pristine KNbO3 possesses an indirect band gap of ~1.576 eV, as valence band maxima (VBM) and conduction band minima (CBM) lie on different symmetry points, i.e., Z and G, of Brillouin zones.However, the calculated band gap is underestimated compared with the experimental value of 3.08 eV [51].Although it is a common feature to undervalue band gaps due to the discontinuity in the exchange-correlation potentials [52], the DFT calculations are expected to be reliable and useful in examining relative changes in electronic band structure and energy band gaps for different KNbO3 models.As seen in Figure 3b,c, the calculated indirect band gaps for Nb K 4• -4V K ′ -KNbO3 and Nb K 4• -K Nb 4′ -KNbO3 models are 1.106 and 1.699 eV, respectively, with VBM and CBM located around the G and A points.Compared to the pristine KNbO3, the conduction band moves downward (upward) to the Fermi level for ), which causes the decrease (increase) in the band gap.Also, the conduction and valence bands in the antisitedefect models become denser, indicating that the electron localization intensifies and could improve the local polarization.To understand the variations in the band structure

Band Structure and Density of States
The band structures along the symmetry directions as well as the total and partial electronic density of states (DOS) are further shown in Figure 3. Figure 3a shows that the pristine KNbO 3 possesses an indirect band gap of ~1.576 eV, as valence band maxima (VBM) and conduction band minima (CBM) lie on different symmetry points, i.e., Z and G, of Brillouin zones.However, the calculated band gap is underestimated compared with the experimental value of 3.08 eV [51].Although it is a common feature to undervalue band gaps due to the discontinuity in the exchange-correlation potentials [52], the DFT calculations are expected to be reliable and useful in examining relative changes in electronic band structure and energy band gaps for different KNbO 3 models.As seen in Figure 3b,c

Ferro-Piezoelectric Properties 3.3.1. Spontaneous Polarization
In modern theory of polarization, spontaneous polarization P s is defined as the difference in polarization between the polar structure of interest and the non-polar (highsymmetry) reference structure.However, for the antisite-defect KNbO 3 , especially Nb 4• K -K 4′ Nb -KNbO 3 , the high-symmetry reference structure cannot be simply determined by inspection.Therefore, linear interpolation is employed to generate intermediate structures of two opposite polar states (P − and P + ), as seen in Figure 4a-c.The direction of polarization along the c axis is defined by the direction of the dipole moment of the oxygen polyhedron.If the Nb cation moves upward relative to the O anion, then it is in the P + configuration; consequently, the P − configuration is oppositely directed and negative.Note that the P − configurations are consistent with the ones in Figure 1a-c.By taking the P − and P + configurations as the start and end points, respectively, the energy as a function of relative off-center displacement of Nb cations for the intermediate structures is shown in Figure 4d-f.As expected, the peak point approximately corresponds to a high-symmetry reference structure that can be utilized to calculate atomic displacement.The migration energy barrier E a for direct P − to P + polarization switching is obtained as 2.1, 6.4, and 27.5 eV for the three models (see Table 3).The larger energy barrier for Nb 4• K -K 4′ Nb -KNbO 3 is probably due to the local structural asymmetry with aggravated distortion and rotation of the oxygen polyhedron (see Figure 1f).Using the Berry phase method [53], the detailed identification of polarization changes is also illustrated in Figure 4g-i.The polarization branches are equally separated by 2P q , namely, P q = eR/V, where P q is the polarization quantum, R is the supercell lattice vector, and V is the volume.For tetragonal KNbO 3 , the magnitude of the spontaneous polarization P s is estimated as 49.1 µC/cm 2 , which matches well with the experimental (37 µC/cm 2 [54]) and previously reported theoretical values (45 µC/cm 2 [55] and 51 µC/cm 2 [56]).More intriguingly, the calculated P s for Nb   d-f) and polarization (g-i) profiles as functions of percentage distortion from the high symmetry (0% distortion) structure to the ferroelectric phase (± 100% distortion) structure using the linear interpolation method.In (g-i), Ps is the spontaneous polarization along the −c axis and Pq is the polarization quantum.The open and solid dots are calculated points and the solid and dashed lines illustrate the evolution along the energy gradient and branches of the polarization lattice, respectively.Notice that the different colored lines represent the various branches of the lattice.
To further elucidate the intrinsic origin of the improved polarization induced by antisite defect complexes , the Ps (see Table 3) is also estimated from the Born effective charge tensor Z * and the corresponding atomic displacement  of all atoms and is computed as: where i denotes the ith atom, and j has {1,2,3} options, corresponding to the a-, b-, and caxis, respectively.In this way, the specific atomic polarization   * can be analyzed by gathering all of the i indices across the same type of atoms in Equation ( 5), as summarized in Table 4.It is clearly shown that the Nb cations contribute most to Ps for KNbO3 (64%) and Nb K 4• -4V K ′ -KNbO3 (73%) due to their relatively larger  33 * compared to K and O atoms, while for Nb K 4• -K Nb 4′ -KNbO3, both Nb and O play a major role in Ps, with   *   ⁄ of 46% and 48%, respectively.In particular, one more Nb (i.e., Nb K   d-f) and polarization (g-i) profiles as functions of percentage distortion from the high symmetry (0% distortion) structure to the ferroelectric phase (±100% distortion) structure using the linear interpolation method.In (g-i), P s is the spontaneous polarization along the −c axis and P q is the polarization quantum.The open and solid dots are calculated points and the solid and dashed lines illustrate the evolution along the energy gradient and branches of the polarization lattice, respectively.Notice that the different colored lines represent the various branches of the lattice.Table 3. Migration energy barrier E a (eV) and spontaneous polarizations along the −c axis P s a and P s b (µC/cm 2 ) using the Berry phase method and Born effective charge, respectively.To further elucidate the intrinsic origin of the improved polarization induced by antisite defect complexes, the P s (see Table 3) is also estimated from the Born effective charge tensor Z * and the corresponding atomic displacement δu of all atoms and is computed as:

Model
where i denotes the ith atom, and j has {1,2,3} options, corresponding to the a-, b-, and c-axis, respectively.In this way, the specific atomic polarization P * s can be analyzed by gathering all of the i indices across the same type of atoms in Equation ( 5), as summarized in Table 4.It is clearly shown that the Nb cations contribute most to P s for KNbO 3 (64%) and Nb   K -K 4′ Nb -KNbO 3 is dominated by the induced dipole moments of O anions in the surrounding cells rather than by the dipole moments of the defects themselves.The enhanced ferroelectricity resulting from the large off-centering of the antisite defects is thus accompanied by geometric asymmetry and severe distortion of oxygen polyhedron polarizing the regions surrounding the antisite pair defects.

Piezoelectric Constants
The elastic stiffness tensor C ij and piezoelectric stress tensor e αj predicted by using the DFPT method for different KNbO 3 models are shown in Tables 5 and 6.The piezoelectric strain tensor d αj is also calculated and is related to e αj and C ij as: where S ij is the elastic compliance tensor and is the inverse matrix of the elastic stiffness tensor C ij .For tetragonal KNbO 3 and Nb 4• K -4V ′ K -KNbO 3 , d 15 , d 31 , and d 33 are the independent piezoelectric strain constants, while for the triclinic Nb 4• K -K 4′ Nb -KNbO 3 , all components are independent (see Equation (S12) in the Supplementary Document) and only 3 of them are summarized for a better illustration.Since our KNbO 3 model in Figure 1a presents the P s along the -c axis due to the downward displacements of Nb cations relative to the O anions, the calculated piezoelectric constants might have the opposite signs as compared with other work.As revealed in Tables 5 and 6, the elastic stiffness constants and piezoelectric constants of KNbO 3 show an overall excellent agreement with previous computational and experimental results [56][57][58][59]61].Especially, d 33 is the most important piezoelectric constant, responding to a uniaxial stress parallel to the spontaneous polarization direction, while the d 31 component is smaller than d 33 since they correspond to the uniaxial stress in a direction orthogonal to the strong Nb-O bonds.This variation can be further confirmed by the considerably smaller S 31 than S 33 (see Equation (S2)).However, the relatively small e 15 and S 55 result in the smallest d 15 , signifying that a miniature change in the in-plane polarization (i.e., P 1 ) is anticipated when a shear stress σ 13 is applied.More intriguingly, as compared to the pristine KNbO 3 , the defected KNbO 3 models exhibit significantly larger piezoelectric strain tensor, which is related to their weakened elastic stiffness.In particular, the Nb 4• K -K 4′ Nb -KNbO 3 presents the d 33 around three times higher than that of KNbO 3 .The incorporation of the antisite pair defects breaks the symmetry of the threefold-degenerate Nb-O bonds and thus effectively stretches the oxygen polyhedron along the c axis due to the amplified S 33 (Equation (S10)).This undoubtedly generates a much larger change in the polarization parallel to the existing spontaneous dipole, P 3 , when subjected to a uniaxial stress σ 33 .Also, the Nb 4• K -K 4′ Nb -KNbO 3 model exhibits relatively large d 13 and d 23 , as associated with the in-plane polarization P 1 and P 2 in the presence of σ 33 , which
, the calculated indirect band gaps for Nb4• K -4V ′ K -KNbO 3 and Nb 4• K -K 4′ Nb -KNbO 3 models are 1.106 and 1.699 eV, respectively, with VBM and CBM located around the G and A points.Compared to the pristine KNbO 3 , the conduction band moves downward (upward) to the Fermi level for Nb 4•K -4V ′ K -KNbO 3 (Nb 4• K -K 4′ Nb -KNbO 3 ), which causes the decrease (increase) in the band gap.Also, the conduction and valence bands in the antisite-defect models become denser, indicating that the electron localization intensifies and could improve the local polarization.To understand the variations in the band structure in detail, the partial DOS for different atoms near the Fermi level, i.e., the energy region from −6 to 6 eV, are presented in Figure3d-f.It is clearly shown that the formation of the upper valance band from −6 to 0 eV for the three models is strongly influenced by O-2p states, followed by Nb-4d states, whereas K-states remain almost neutral.In the lower conduction band, above the Fermi level between 0 and 6 eV, the 4d-states of Nb along with partial 2p-states of O play a major role.The hybridization between Nb-4d and O-2p orbitals is thus dominant for the band gaps.As compared to the pristine KNbO 3 , the weaker p-d hybridization induced by the four K vacancies narrows the band gap for Nb 4• K -4V ′ K -KNbO 3 , whereas the antisite pair defects strengthen p-d covalent interactions, leading to an increasing band gap for Nb 4• K -K 4′ Nb -KNbO 3 .indetail, the partial DOS for different atoms near the Fermi level, i.e., the energy region from −6 to 6 eV, are presented in Figure3d-f.It is clearly shown that the formation of the upper valance band from −6 to 0 eV for the three models is strongly influenced by O-2p states, followed by Nb-4d states, whereas K-states remain almost neutral.In the lower conduction band, above the Fermi level between 0 and 6 eV, the 4d-states of Nb along with partial 2p-states of O play a major role.The hybridization between Nb-4d and O-2p orbitals is thus dominant for the band gaps.As compared to the pristine KNbO3, the weaker pd hybridization induced by the four K vacancies narrows the band gap for Nb K 4• -4V K ′ -KNbO3, whereas the antisite pair defects strengthen p-d covalent interactions, leading to an increasing band gap for Nb K 4• -K Nb 4′ -KNbO3.

Figure 3 .Figure 3 .
Figure 3. Energy band structures and the total and partial density of states (DOS) for different elements and orbitals in (a,d) KNbO3, (b,e) Nb K 4• -4V K ′ -KNbO3, and (c,f) Nb K 4• -K Nb 4′ -KNbO3.The black Figure 3. Energy band structures and the total and partial density of states (DOS) for different elements and orbitals in (a,d) KNbO 3 , (b,e) Nb 4• K -4V ′ K -KNbO 3 , and (c,f) Nb 4• K -K 4′ Nb -KNbO 3 .The black dashed lines are the Fermi level, set to zero energy.The valance band maximum and the conduction band minimum are depicted by the blue and red lines, respectively, in (a-c).

Figure 4 .
Figure 4. Atomic structures of pathways corresponding to the polarization switching of (a) KNbO3, (b) Nb K 4• -4V K ′ -KNbO3, and (c) Nb K 4• -K Nb 4′ -KNbO3, and their corresponding energy (d-f) and polarization (g-i) profiles as functions of percentage distortion from the high symmetry (0% distortion) structure to the ferroelectric phase (± 100% distortion) structure using the linear interpolation method.In (g-i), Ps is the spontaneous polarization along the −c axis and Pq is the polarization quantum.The open and solid dots are calculated points and the solid and dashed lines illustrate the evolution along the energy gradient and branches of the polarization lattice, respectively.Notice that the different colored lines represent the various branches of the lattice.

Figure 4 .
Figure 4. Atomic structures of pathways corresponding to the polarization switching of (a) KNbO 3 , (b) Nb 4• K -4V ′ K -KNbO 3 , and (c) Nb 4• K -K 4′ Nb -KNbO 3 , and their corresponding energy (d-f) and polarization (g-i) profiles as functions of percentage distortion from the high symmetry (0% distortion) structure to the ferroelectric phase (±100% distortion) structure using the linear interpolation method.In (g-i), P s is the spontaneous polarization along the −c axis and P q is the polarization quantum.The open and solid dots are calculated points and the solid and dashed lines illustrate the evolution along the energy gradient and branches of the polarization lattice, respectively.Notice that the different colored lines represent the various branches of the lattice.

Table 4 .
Averaged Born effective charge Z * , atomic displacement δu (Å), the effective atomic polarization along the −c axis P * s (µC/cm 2 ), and its corresponding percentage contribution P * s /P s for different types of atoms in the pristine and defected KNbO 3 models.The oxygen atoms are divided into four types: O Nb and O K , representing the nearest-neighboring O atoms located around anti-site defects Nb 4• K and K 4′ Nb ; and O I (collinear with Nb along the c axis) and O II (coplanar with the Nb in the ab plane), highlighting the structural difference between the two O sites.

Model L a , L b L c α β γ Volume Energy Space Group
KNbO 3 are 65.5 and 68.2 µC/cm 2 , respectively, considerably larger than that of the pristine KNbO 3 .
giving rise to additional   * of 10 µC/cm 2 with a considerably high  3 of 0.87 Å, which is responsible for the improved Ps.On the other hand, despite antisite defects, Nb K 4• and K Nb 4′ produce more   * of 9.35µC/cm 2 , and the total polarization contribution from Nb and K is comparable to that of KNbO3.Consequently, the overall largest Ps in Nb K 4• -K Nb 4′ -KNbO3 is dominated by the induced dipole moments of O anions in the surrounding cells rather than by the dipole moments of the defects themselves.The enhanced s of 10 µC/cm 2 with a considerably high δu 3 of 0.87 Å, which is responsible for the improved P s .On the other hand, despite antisite defects, Nb 4• K and K 4′ Nb produce more P * s of 9.35µC/cm 2 , and the total polarization contribution from Nb and K is comparable to that of KNbO 3 .Consequently, the overall largest P s in Nb * 4

Table 6 .
[61]oelectric constants (C/m 2 for e ij and pC/N for d ij ) for different KNbO 3 models.Experimental result for tetragonal KNbO 3[61].bThecalculated piezoelectric stress constants e ij are derived based on C ij and d ij from each reference. a